No it doesn't let you know. It's something I hope they add to the next game. Disney magical World says you own it.
Do you want to buy another?
Use the catalog machine to check if you have it.
Are you asking about getting them from KK? It doesn't let you know and I don't think the shop is open at that hour. You can just ask for a song you know you don't have before requesting any other songs or just keep track of which ones you already have.
If you're asking about buying them from the shop, it's the same as other items. It doesn't let you know so you have to check if you've catalogued them.
However you obtain your music, it won't let you know if you already have it.
However, if you have multiple characters then I highly recommend transferring any duplicates over to them so you can try to max out your record collection stored in the music players on up to all four characters! Really handy for if you want to have multiple houses and to add music to each room.